Question

Show that $f_{x y z}=f_{y z x}=f_{z x y}$ for $$ f=x y e^{y z} $$

   Show that $f_{x y z}=f_{y z x}=f_{z x y}$ for
$$
f=x y e^{y z}
$$

The Chemistry Maths Book

The Chemistry Maths Book

Erich Steiner 2nd Edition

Chapter 9, Problem 16

Instant Answer

Solved by Expert John Nicolle

01/03/2022

Step 1

The partial derivative of $f$ with respect to $x$ is: $$f_x = y e^{yz}$$ Show more…

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Show that $f_{x y z}=f_{y z x}=f_{z x y}$ for $$ f=x y e^{y z} $$

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Key Concepts

Chain Rule for Multivariable Functions

The chain rule in multivariable calculus is used for differentiating composite functions. When a function depends on one or more other variables that are themselves functions of additional variables, the chain rule provides a systematic way to compute the derivative, accounting for the dependencies between these variables.

Mixed Partial Derivatives

This concept refers to the multiple derivatives that are taken with respect to different variables in a multivariable function. Mixed partial derivatives involve differentiating with respect to one variable and then another. The key aspect is that, under appropriate conditions, the order of taking these derivatives does not affect the result, allowing for symmetry in the computation.

Clairaut's Theorem

Clairaut's theorem (also known as Schwarz's theorem) states that if the mixed partial derivatives of a function are continuous around a given point, then the order of differentiation does not matter. This means that the value of the mixed partial derivative remains the same regardless of the order in which the differentiations are performed.

Product Rule for Differentiation

The product rule is a fundamental principle used when differentiating functions that are products of two or more functions. In the context of multivariable calculus, this rule is applied by differentiating one function while keeping others constant, and then summing the results. It is essential for computing derivatives of functions where variables are multiplied together.

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